Question 3 on the OCR MEI C3 June 2015 paper. Find the exact value of Integral x^3 ln x dx between 1 and 2.

This is a good candidate for integration by parts because it is a product of two functions – ln(x) and x^3, and x^3 is easy to integrate. The definition for integration by parts is: integral(u dv/dx) dx = uv – integral(v du/dx) dx. We choose u = ln(x) as ln(x) is difficult to integrate and dv/dx = x^3. This means v = (x^4)/4 and du/dx = 1/x. Integral(ln(x) x^3) dx = ln(x) * (x^4)/4 – integral(((x^4)/4 )* 1/x) dx = ln(x) * (x^4)/4 - integral((x^3)/4 ) dx = ln(x) * (x^4)/4 – (x^4)/16. Now we must substitute in the bounds. [ln(2) * (2^4)/4 - (2^4)/16] - [ln(1) * (1^4)/4 - (1^4)/16 ]. Remember that ln(1) = 0 giving an answer of ln(2) * 16/4 - 16/16 + 1/16 = 4 ln(2) – 15/16.